Mixed-Type Reverse Order Laws Associated to $\{1,3,4\}$-Inverse

DOI：10.3770/j.issn:2095-2651.2019.05.009

 作者 单位 张海燕 商丘师范学院数学与统计学院, 河南 商丘 476000 邓春源 华南师范大学数学科学学院, 广东 广州 510631

本文主要研究算子$A$, $B$和$AB$值域闭时, $\{1,3,4\}$-逆的混合逆序律问题. 给出$B\{1,3,4\}A\{1,3,4\}\subseteq (AB)\{1,3\}$ 当且仅当 $R(A^*AB)\subseteq R(B)$. 并且对任意的 $A^{(134)}\in A\{1,3,4\}$, 有 $(A^{(134)}AB)\{1,3,4\}A\{1,3,4\}= (AB)\{1,3,4\}$ 当且仅当 $R(AA^*AB)\subseteq R(AB).$作为上述结果的应用, 建立关于 Moore-Penrose 逆和 $\{1,3,4\}$-逆的混合逆序的新刻画.

In this paper, we study the mixed-type reverse order laws to $\{1,3,4\}$-inverses for closed range operators $A$, $B$ and $AB$. It is shown that $B\{1,3,4\}A\{1,3,4\}\subseteq (AB)\{1,3\}$ if and only if $R(A^*AB)\subseteq R(B)$. For every $A^{(134)}\in A\{1,3,4\}$, it has $(A^{(134)}AB)\{1,3,4\}A\{1,3,4\}= (AB)\{1,3,4\}$ if and only if $R(AA^*AB)\subseteq R(AB)$. As an application of our results, some new characterizations of the mixed-type reverse order laws associated to the Moore-Penrose inverse and the $\{1,3,4\}$-inverse are established.